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  2. Critical point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Critical_point_(mathematics)

    A critical value is the image under f of a critical point. These concepts may be visualized through the graph of f: at a critical point, the graph has a horizontal tangent if one can be assigned at all. Notice how, for a differentiable function, critical point is the same as stationary point.

  3. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.

  4. Derivative test - Wikipedia

    en.wikipedia.org/wiki/Derivative_test

    After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. [1] If the function f is twice-differentiable at a critical point x (i.e. a point where f ′ (x) = 0), then:

  5. Newton's method in optimization - Wikipedia

    en.wikipedia.org/wiki/Newton's_method_in...

    The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.

  6. Second derivative - Wikipedia

    en.wikipedia.org/wiki/Second_derivative

    The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.

  7. Maximum and minimum - Wikipedia

    en.wikipedia.org/wiki/Maximum_and_minimum

    Although the first derivative (3x 2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) Unique global maximum at x = e. (See figure at right) xx: Unique global maximum over the positive real numbers at x = 1/e. x 3 /3 − x: First derivative x 2 − 1 and second derivative 2x.

  8. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several

  9. Phase line (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Phase_line_(mathematics)

    A line, usually vertical, represents an interval of the domain of the derivative.The critical points (i.e., roots of the derivative , points such that () =) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an ...